A060319 -id:A060319 - cp's OEIS frontend

A060384 Number of decimal digits in n-th Fibonacci number.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Labos Elemer, Apr 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].
a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

Original entry on oeis.org

1, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, 450, 432, 552, 360, 420, 690, 504, 880, 630, 600, 756, 720, 900, 792, 840, 1296, 1050, 1350, 1140, 1080, 1200, 1824, 1260, 1512, 1320, 1560, 1680

Offset: 0

Labos Elemer, Mar 28 2001

From Jon E. Schoenfield, Dec 28 2016: (Start)
 Note that the presence of incompletely factored Fibonacci numbers with indices as low as 1301 does not prevent the drawing of conclusions such as "a(44) = 1320" with certainly. Using F(1301) as an example, the compact table of Fibonacci results at the Kelly site indicates that F(1301) = p*q*r*c where p=6400921, q=14225131397, r=100794731109596201, and c is a 238-digit unfactored composite number. The complete factorization of every Fibonacci number up to F(1000) is explicitly given elsewhere on the site, and those results allow quick verification that a(n) <= 900 for all n in [0..34], so 1301 cannot be a term unless F(1301) has at least 35 distinct prime factors, which would require c to have at least 32 distinct prime factors, at least one of which would have to be less than ceiling(c^(1/32)) = 26570323, but trial division of c by every prime less than 26570323 shows that c has no prime factors that small. Thus, while A022307(1301) is unknown, it is certain that 1301 is not a term in this sequence. Similarly, making use of known factors, it can be proved that F(n) cannot have 44 or more distinct prime factors for any n < 1320, so since F(1320) has exactly 44 distinct prime factors, it is established that a(44) = 1320. (End)
a(47) >= 2835, a(48..68) = (2040, 1800, 2736, 2730, 1890, 1980, 2520, 2280, 2100, 2160, 2640, 3300, 3060, 3150, 2520, 3120, 3696, 3240, 3990, 3360, 3420), a(69) >= 4400, a(75) = 4320, a(77) = 4200, a(79) = 3780. - Max Alekseyev, Feb 03 2025

			n=9: F(80) = 23416728348467685 = 3 * 5 * 7 * 11 * 41 * 47 * 1601 * 2161 * 3041.
n=25: F(690) = 2^3 * 5 * 11 * 31 * 61 * 137 * 139 * 461 * 691 * 829 * 1151 * 1381 * 4831 * 5981 * 18077 * 28657 * 186301 * 324301 * 686551 * 1485571 * 4641631 * 117169733521 * 2441738887963981 * 3490125311294161 * 25013864044961447973152814604981 is the smallest Fibonacci number with exactly 25 distinct prime factors.

Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
Hisanori Mishima, Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).

Cf. A001605, A005478, A022307, A051694, A060319.

Row n=1 of A303217.

First /@ SortBy[#, Last] &@ Map[First@ # &, Values@ GroupBy[#, Last]] &@ Table[{n - Boole[n == 2], #, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 300}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)
Module[{ff=Table[{n,PrimeNu[Fibonacci[n]]},{n,1400}]},Table[ SelectFirst[ ff,#[[2]]==k&],{k,0,40}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 28 2018 *)

my(o=[],s); print1(1); for(n=1,20, s=0; until( o[s]==n, #o

a(n) = min (k : A022307(k) = n).

Corrected by Shyam Sunder Gupta, Jul 20 2002
Edited by M. F. Hasler, Nov 01 2012
a(35)-a(40), a(42), a(44) computed based on Kelly's data in A022307 by Jon E. Schoenfield, Dec 28 2016
a(41), a(43), a(45)-a(46) from Max Alekseyev, Feb 03 2025

A060385 Largest prime factor of n-th Fibonacci number.

Original entry on oeis.org

2, 3, 5, 2, 13, 7, 17, 11, 89, 3, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 61, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 421, 433494437, 307, 109441, 28657, 2971215073, 1103

Offset: 3

Labos Elemer, Apr 03 2001

nonn

For n > 12, Fibonacci(n) is divisible by a primitive prime factor (one not dividing Fibonacci(1), ..., Fibonacci(n-1)). But all primes up to n-2 divide smaller Fibonacci numbers, see A001602, so a(n) >= n-1 for n > 12. This strengthens a more general theorem of Bravo and Luca. - Charles R Greathouse IV, Feb 01 2013

			F(82) = 2789 * 59369 * 370248451, so a(82) = 370248451.

Tyler Busby, Table of n, a(n) for n = 3..1422 (terms 3..1000 from Charles R Greathouse IV, terms 1001..1408 from Amiram Eldar)
U. Alfred, On the form of primitive factors of Fibonacci numbers, Fibonacci Quarterly 1:1 (1963), pp. 43-45.
Jhon J. Bravo and Florian Luca, On the largest prime factor of the k-Fibonacci numbers, arXiv:1210.4101 [math.NT], 2012.
D. E. Daykin and L. A. G. Dresel, Factorization of Fibonacci numbers, Fibonacci Quarterly 8:1 (1970), pp. 23-30.

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A193615.

[Maximum(PrimeDivisors(Fibonacci(n))): n in [3..50]]; // Vincenzo Librandi, Dec 25 2016

Table[First[Last[FactorInteger[Fibonacci[n]]]], {n, 3, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

a(n)=my(f=factor(fibonacci(n))[,1]);f[#f] \\ Charles R Greathouse IV, Feb 01 2013

a(n) >= n - 1 for n > 12, see comments. It is not hard to show that a(n) > 1000 for n > 88. Similarly a(n) > 20641 for n > 120. - Charles R Greathouse IV, Feb 01 2013

A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 13, 3, 2, 5, 89, 2, 233, 13, 2, 3, 1597, 2, 37, 3, 2, 89, 28657, 2, 5, 233, 2, 3, 514229, 2, 557, 3, 2, 1597, 5, 2, 73, 37, 2, 3, 2789, 2, 433494437, 3, 2, 139, 2971215073, 2, 13, 5, 2, 3, 953, 2, 5, 3, 2, 59, 353, 2, 4513, 557, 2, 3, 5, 2, 269, 3, 2, 5

Offset: 1

Labos Elemer, Apr 03 2001

nonn

			For n=82: F(82) = 2789*59369*370248451, so a(82)=2789.

Tyler Busby, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Alois P. Heinz, derived from Kelly's data, terms 1001..1408 from Amiram Eldar)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A139044.

[1,1] cat [Minimum(PrimeDivisors(Fibonacci(n))): n in [3..70]]; // Vincenzo Librandi, Dec 25 2016

f[n_] := (FactorInteger@ Fibonacci@ n)[[1,1]]; Array[f, 70] (* Robert G. Wilson v, Jul 07 2007 *)

a(n) = if ((f=fibonacci(n))==1, 1, factor(f)[1,1]); \\ Michel Marcus, Nov 15 2014

a(n) = A020639(A000045(n)). - Michel Marcus, Nov 15 2014

Better definition from Omar E. Pol, Apr 25 2008

A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 21, 3, 610, 34, 5, 6765, 987, 55, 8, 832040, 46368, 2584, 144, 13, 102334155, 14930352, 196418, 10946, 377, 89, 190392490709135, 4807526976, 267914296, 317811, 3524578, 4181, 233, 1548008755920, 37889062373143906, 86267571272, 701408733, 2178309, 9227465, 17711, 1597

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   2,   21,     610,        6765,      832040,        102334155, ...
   3,   34,     987,       46368,    14930352,       4807526976, ...
   5,   55,    2584,      196418,   267914296,      86267571272, ...
   8,  144,   10946,      317811,   701408733,     225851433717, ...
  13,  377, 3524578,     2178309,  1134903170,   10610209857723, ...
  89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Column k=3 gives A137563.
Row n=1 gives: A060319.
Cf. A000045, A001221, A303216, A303217.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

nmax = 12(*rows*);
maxIndex = 200; (* increase if message "part does not exist" *)
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k &];
T = Array[col, nmax];
A[n_, k_] := Fibonacci[T[[k, n]]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2021 *)

A(n,k) = A000045(A303217(n,k)).

A001221(A(n,k)) = k.

A359848 a(n) is the smallest tribonacci number (A000073) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 24, 504, 35890, 8646064, 1697490356184, 120879712950776, 98079530178586034536500564, 748829299860308729347600, 119816209721856219780831547518850, 15418262617564622254988364568360573618470100684551892712710640455037970

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

nonn

			a(4) = 35890, because 35890 is a tribonacci number with 4 distinct prime factors {2, 5, 37, 97} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tribonacci Number

Cf. A000073, A001221, A060319, A359849, A359850.

a(11) from Daniel Suteu, Jan 17 2023

A359849 a(n) is the smallest tetranacci number (A000078) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 15, 1490, 39648, 28074040, 100808458960497, 9966792788887776, 4997150614173857218560, 1835682610171974487231869, 889487735339682550112673527109223032, 52499930084496170026238596234557616056408988199026780675759699719704592

Offset: 0

Ilya Gutkovskiy, Jan 15 2023

nonn

			a(4) = 39648, because 39648 is a tetranacci number with 4 distinct prime factors {2, 3, 7, 59} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Tetranacci Number

Cf. A000078, A001221, A060319, A359848, A359851.

a(n) = A000078(A359851(n)). - Daniel Suteu, Jan 18 2023

a(11) from Daniel Suteu, Jan 18 2023

A359960 Smallest Niven (or Harshad) number (A005349) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 690690, 14804790, 223092870, 8254436190, 200560490130, 8222980095330, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1987938667108592728530, 117288381359406970983270, 7858321551080267055879090

Offset: 0

Bernard Schott, Jan 20 2023

a(11) = 200560490130; a(13) = 304250263527210.
a(n) >= A002110(n) = prime(n)#.
Many terms are primorial numbers, see A360011.

			2310 = 2*3*5*7*11 is the smallest integer with 5 prime factors because it is a primorial number, as 2310 / (2+3+1+0) = 385, 2310 is a Niven number: a(5) = 2310.

Giovanni Resta, Harshad numbers.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A002110, A005349, A113315, A360011.

Similar: A060319 (Fibonacci), A083002 (oblong), A359961 (Zuckerman).

a(n) = my(k=1); while ((k % sumdigits(k)) || (omega(k) != n), k++); k; \\ Michel Marcus, Jan 20 2023

omega_niven(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && v%sumdigits(v) == 0, listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_niven(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jan 22 2023

a(8)-a(9) from Michel Marcus, Jan 20 2023

a(10)-a(19) from Daniel Suteu, Jan 22 2023

A229490 The smallest Lucas number having exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 18, 322, 1860498, 599074578, 2537720636, 4721424167835364, 215002084978043708894524818, 8784200221406821330636, 3739702405897758836585154759070350989, 16342986943522226847837781364

Offset: 0

T. D. Noe, Oct 28 2013

nonn

T. D. Noe, Table of n, a(n) for n = 0..22

Cf. A000032 (Lucas numbers).
Cf. A060319 (corresponding sequence for Fibonacci numbers).
Cf. A229491 (indices of these Lucas numbers).

A359852 a(n) is the smallest Fibonacci n-step number with exactly n distinct prime factors.

Original entry on oeis.org

21, 504, 39648, 6930, 12669125245488, 471771076278370, 32818036405994618064, 71577732779401085355729600, 204945946670840805166309694624676331385919836360545974559162291811394735721440

Offset: 2

Ilya Gutkovskiy, Jan 15 2023

nonn

			a(3) = 504, because 504 is a tribonacci number with 3 distinct prime factors {2, 3, 7} and this is the smallest such number.

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A001221, A060319, A359848, A359849, A359853.

a(n) = my(v=vector(n+1), x=1, y=n+1); v[1]=v[y]=1; while(omega(v[x])!=n, y=x; x=x%(n+1)+1; v[x]=2*v[y]-v[x]); v[x]; \\ Jinyuan Wang, Jan 16 2023

a(10) from Jinyuan Wang, Jan 16 2023

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A060384 Number of decimal digits in n-th Fibonacci number.

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A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

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A060385 Largest prime factor of n-th Fibonacci number.

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A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

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A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A359848 a(n) is the smallest tribonacci number (A000073) with exactly n distinct prime factors.

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A359849 a(n) is the smallest tetranacci number (A000078) with exactly n distinct prime factors.

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